13 March 2008

Streamlined math curriculum

"Streamlining" in this context seems to mean covering fewer topics in each grade, but covering them more thoroughly; I'm not sure if it means covering less topics overall. The article also tells us that "The report tries to put to rest the long and heated debate over math teaching methods" -- somehow I don't think one report can do that, since people have been debating whether the teacher's role is to hand down facts from on high or help students discover things on their own from time immemorial. (And this isn't unique to mathematics.)

Larry Faulkner, who chaired the panel, says that the “talent-driven approach to math, that either you can do it or you can’t, like playing the violin” needs to be changed; this is something I agree with. You don't see people just saying "I can't read" and throwing up their hands in disgust -- okay, maybe you do, but not nearly as many as you do with math. I realize that this comparison isn't fair -- reading really is more fundamental than other kinds of learning, if only for the reason that most other learning involves reading -- but I'm making it anyway.

The panel also advocates shorter textbooks. As an instructor of calculus courses, I support this; the report isn't talking about college texts but one would hope that such a thing might filter upwards. Then I could actually bring the text home. I try not to with our current text, which is 1368 pages, because I commute on foot and so carrying around extra pounds is a Bad Thing. (By the way, reading the customer-written reviews of textbooks on Amazon is kind of funny in a depressing way; one gets the sense that students are often taking out their frustrations at their instructors on the textbook.)

10 comments:

David
said...

$180.05 on Amazon. I'm gagging right now.

Calculus would go a whole lot faster (careful, wild speculation coming) if you spent the first semester really making clear the difference notions of limits and continuity, and convergence of sequences of series. Get the hard numerical stuff out of the way. Then you could spend the next two semesters doing the arty geometry part of 'analytic geometry' (IIRC, my calculus book was called 'Calculus with Analytic Geometry', but that was, uhh, 17 years ago).

that price is a bit misleading. The 5th edition is actually not the current edition, although it's the one I know well; there's a sixth edition, which is $129. Still way more expensive than these books have any right to be, though.

But here we use that textbook for three semesters. A part of me wonders -- if the book were split into a series of three books, what would the price of each book be? Probably significantly more than $43...

And the problem with your speculation is that the calculus sequence has to be structured in such a way that people can take the first semester, or the first two semesters, and come away with something that's useful for them in Whatever It Is They Do. The primary audience of calculus courses is not future mathematicians, but future engineers and businesspeople, and they don't "need" the rigorous notions of limits and convergence and so on.

The other issue with getting a solid grounding in limits and continuity in a freshman calculus class is that for most students, that will only make it more confusing. Many of the students I tutor still struggle with their algebra skills, and they don't even have an intuition for how they can manipulate an equation to solve for some variable, much less how the equation they write relates to a graph they can draw. I think most of my students would cry if I were to start throwing epsilons and deltas at them.

Now, if math education at the elementary and high school level were improved to the point where students could gain a strong intuition for both algebraic and arithmetic skills AND a reasonable understanding of rigorous proof, then maybe it would be possible to teach limits and continuity in a rigorous way. For now I think it's better to teach the intuition that "a continuous function is one I can draw without lifting my pencil from the paper".

Where are the word problems on this "streamlined" curriculum? It continues to treat students as computers on which such-and-such software must be installed. The same nauseating, mind-numbing trash! No wonder most Americans don't know the difference between mathematics and accounting!

Limits and continuity are just mathematical fashion fads to make calculus "rigorous," they are totally unnecessary and useless for most computations. When will you learn to see the difference between mathematical ideology and the practical aspects of the subject?

if the book were split into a series of three books, what would the price of each book be? Probably significantly more than $43...

Actually, no. Yale had a deal worked out with Stewart's publisher where we'd assure a certain number of students would be in each course each semester. Then they ran off three books for us. Each was just the relevant sections of Stewart 6, like the "MATH120" book started at chapter 12 and went through 16, since we didn't do the differential equations stuff in 17.

The books were softcover, and ran about $20-$25 each. Students only had to buy the one for the current course, so if a major version changed between now and next semester, they didn't have to buy a whole new text.

Actually, that one's not coming into play, because when Stewart 7 comes out, Yale is buying access to it online. Students can print it out at Kinko's if they really want, but reading it online (and doing automatically-graded homework online!) will be free.

Psst.. You know it's me, John, right? Blogger just finally got on board with OpenID, so I can use it to leave comments, but it leaves them with the third-level domain name. You don't have to use that when you respond ;)

I'll spare you the long winded reply [I've tried twice to make it concise, but I ramble], and simply note that I disagree that it is a problem of learning outcomes, but rather a problem of orthodox expectations [If you don't know whatever-it-is-that-they-do, it's hard to teach them something-that-is-useful]

The rigorous notion of limits, continuity, and convergence are exactly suited to engineers and businesspeople. Those things are the canonical end to a discussion of approximation. There is no end to problems that ask, "How close is close enough?"

By the by, my niece frantically asked me for help with her trigonometry homework Sunday night, as I was getting ready to leave my in-laws. It is a travesty what passes for high school math. Math tells a story, and the teachers and teaching the kids how to read it. They go from:

graph y = sin(3x + 7π/11) - 2 -TO-You are riding on a ferris wheel, at .3 of s, you are 37 feet off the ground, at the top of the ferris wheel you are 43 feet of the ground, and you are at the bottom of the ferris wheel after ... Solve.